During a bandwidth-meter development project applicant's company determined that certain products (e.g., an ELS-7000 WSM wavemeter) has insufficient pixel sampling resolution when used at certain wavelengths. This insufficiency results in certain measurement errors, a root cause of which is an aliasing effect of the etalon spectrometer fringe image against the periodic grid of detector elements (e.g., a linear array of photodiodes). To reduce the level of certain measurement errors connected with undersampling of the spectrometer fringe image applicants propose certain modifications to wavemeter technology. Such wavemeter technology may be utilized, e.g., in measuring the spectral purity of line narrowed DUV laser system output laser light pulses, e.g., in excimer and/or molecular fluorine gas discharge lasers.
JenaOptik (Zeiss) uses a similar technique in their CCD cameras for microscopy (ProgRes products).
Ben-Esra, et al., “Jitter camera: high resolution video from a low resolution detector,” IEE (2000) and Wuttig, et al., “Subpixel analysis of double array grating spectrometer,” published in Descour/Schen, Imaging Spectrometry VII, Proc. SPIE, Vol. 4480 (2002) pp. 334-344 discuss aspects of image resolution techniques aspects of which may be useful according to aspects of embodiments of the present invention, the disclosures of which are hereby incorporated by reference.
To better understand aspects of the problems associated with under-sampling and aliasing in the present context, reference is made to FIG. 5. In FIG. 5a there is illustrated an example of a linear array 100 of photodiodes each representing an imaging pixel and a pixel in the output image of an example of a known center wavelength/bandwidth detector, such as those referenced in the patents and applications noted above. The pixels are labeled from convenience of reference X, x+1, x+2, x+3, x+4 and x+5. Assuming that an image of a light source represented by two square waves, A and B, as shown in FIG. 5b, of photonic energy are incident upon the array 100 so as to just illuminate two pixels x+2 and x+3, i.e., there is no spillover into the adjacent pixels x+1 on the left side or x+4 on the right side sufficient to register any detected intensity during some intensity sensing integration period. As shown in FIG. 5e the output of the detector using typical intensity sensing circuitry could be interpreted as the very same two square wave pulses 1 and 1′ shown outlined by solid lines in FIG. 5e. 
Turning to FIG. 5c, there is represented the same two square wave pulses A and B spatially phase shifted by about ½ a pixel width. In this event, the integrated intensity out of pixel x+2 will only be one half of what it was in the first example noted above, as illustrated in FIG. 5e by a dashed line defining an output 2. the output for pixel x+3 will remain the same, as indicated by the dashed line for output 2′, and now pixel x+4 will have an output where it had none before, at a level of about ½ that of the output of pixel X+3. The output of the detector for such an input integrated by the photodiodes X+2, X+3 and X+4 in the linear array 100 could be interpreted as a saw tooth shaped image as denoted by the dashed line triangle in FIG. 5e. 
Finally for an alignment as illustrated in FIG. 5d, the illumination of pixel x+2 is reduced to about 30% of what it was in the alignment of FIG. 5b and the illumination of pixel x+4 is about 70% of the illumination of the pixel x+3. These are represented by the dotted line outlines of the intensities for the pixels x+2 and x+5 and the dotted line triangle of FIG. 5e. 
A similar version of these phenomena cause the differences in the outputs illustrated by way of examples in FIGS. 2 and 3. A very simplified version of the sampling of an interference fringe image of the type shown in FIGS. 2 and 3 is shown schematically for illustrative purposes in FIG. 6. FIG. 6 illustrates in a very simplified fashion the sampling of a fringe from an interference pattern in a typical known bandwidth detector. The representative pixels in a linear array of a photodetector are again labeled X through x+5 for convenience. As shown by way of illustration the points on the intensity curve for the fringe, e.g., points a, B and C, e.g., on one side of a peak of the intensity of the fringe pattern may be sampled by pixels X, x+1 and x+2. These may on average integrate an intensity that indicates approximately the respective points on the curve of the fringe peak.
Several problems with under-sampling are illustrated in FIG. 6 illustratively and schematically and not to scale. First, the max of the fringe peak may be determined to be either C or D, or both if they are equal in intensity which the real maximum is labeled MAX. Thus determining some threshold intensity bandwidth measurement, e.g., full width at 80% maximum (“FW80% M”) may occur at the intensity value labeled FW80% M′ in FIG. 6 because the real peak is not detected. further the error between where the real peak is and the detected peak can be seen to vary as the number of pixels in relation to the field covered by the image of the fringe increases.
Some prior art wavemeters have employed peak estimating algorithms, e.g., taking, by way of example the values for pixels x+1, x+2 and X+3, points B, C and D on the fringe curve and doing a parabolic estimate of the actual shape of the curve to derive something closer to the actual intensity maximum MAX, which, as illustrated can also be in error, e.g., being determined by such a parabolic estimation algorithm to be INT MAX. Further, as shown illustratively and schematically and not to scale, in FIG. 7, assuming the detector circuitry determines that INT MAX is very close to the actual MAX, the nature of the detector output reading with under-sampled spatial distribution of pixels in the linear array can result in an error in detecting the width at the selected threshold intensity. Typical known algorithms for determining the width at a selected FWX % M, e.g., FW80% M use an interpolation between sampled points on the fringe curve, e.g., points B and C. the interpolation may be linear, as shown in FIG. 7, in which event, wherever along the line between point B and point C where it is determined by the readout electronics that the threshold intensity occurs, e.g., 80% of the max, in addition to the possible errors cause by the limitations of the finite pixel array noted above, the point on the interpolation line is separated from the actual point on the curve by an error ΔFWXM. As can be seen from the exemplary representation of FIG. 7, if the number of pixels is doubled (the pixel width halved) this error can be reduced, but still exists.
These errors can be exacerbated by the impact of the undersampling using a finite pixel count and by movement of the image on the linear array grid of photodetectors, as noted above.
Some prior art wavemeters have employed more sophisticated interpolation algorithms, e.g., using a polynomial, e.g., Ax2 +bx to simulate the curve between the sampled points, e.g., B and C, to beer estimate the width of the fringe intensity curve at the point along the curve at which the threshold occurs, but even these are subject to errors of the type noted above.
To add to this, the width of the fringe intensity curves varies (gets narrower) at the extremities of the fringe pattern being sampled, so that, as shown by way of example in FIG. 8. applicants have found, s illustrated in FIG. 8 that, e.g., along a linear array of photodiodes used to sample the fringe intensities of a given fringe, the number of pixels illuminated decreases. Thus, the apparent bandwidth distance, e.g., at some threshold, e.g., FW20% M increases in the fringes closer to one extremity of the linear array 100. Applicants have also found that this trend is somewhat tolerable and can be determined for a given wavemeter and corrected for by subtracting an amount determined from the trend shown in FIG. 8 (some algorithms simulate the actual trend curve and some use a simple linear correction which in older system may have sufficed. However, even with such correction applicants have found that the above noted errors due in part to undersampling and tie resultant aliasing as well as some other adverse impacts some of which may not be fully appreciated or understood at this time by applicants cause the sever fluctuations in apparent fringe width shown by the lighter graphical representations in FIG. 8.
Applicants propose certain improvements to sampling and readout of fringe widths at the desired threshold values as noted below.